The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 641 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 641

f(t,t) = it

Integrating the first relationship we obtain

u( ,

Λu = e∫t fts )ds

t

Now suppose that in the interval u∈(0,T] the forward rate obeys the following SDE:

df= α(t,u)dt+ σ(t,u)dBt

Equivalently, this means that for each u∈(0,T] the following relationship holds:

t t

ftu ( , )= f( 0 ,u)+ ∫ α(su, )ds+ ∫σ(su, )dBˆs

0 0

Stochastic differentiation yields

u u

d–∫fts ( , )ds = ftt ( , )dt+ ∫dts f( , )ds

t t u

= it()dt– ∫[α(ts, )dt+ σ(ts, )dBˆ t]ds

t = it()dt– α*(tu, )dt+ σ*(tu, )dBˆt

where

u

α*(tu, )= ∫ α(ts, )ds

t

u

σ*(tu, )= ∫ σ(ts, )ds

t

Using Itô’s lemma, it can be demonstrated that the term structure process obeys the following SDE: